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For a sinusoidal mean jet on an $f$ plane, its QG potential-vorticity gradient is $({\pi }^2/L^2+k_d^2)U$. Ground-stationary modes satisfy the displayed circle and can solve the full stationary shear equation even though their scale does not meet a local short-wave assumption. The local stationary group vector is $2Uk(k,l)/(k^2+l^2+k_d^2)$, reducing to a zonal scalar formula only for $l=0$.

At $\chi ={\chi }^w{\delta }^{-1}$, a determinant-character twist reduces to $\chi (\mathrm{diag}(a,d))=|a|$. In the open Bruhat coordinate, a section $g_0(x)=1$ for $|x|\le 1$ and $g_0(x)=|x{|}^{-1}$ otherwise has $\pi (\mathrm{diag}(\varpi ,1))g_0-q^{-1}{g}_0=(1-q^{-1})1_{\varpi \mathfrak{o}}$. Its additive Haar integral is nonzero, whereas every unipotent translation difference has zero such integral. The difference therefore survives in the Jacquet module, proving that this two-dimensional self-extension is nonsplit and has just one quotient functional of the common linear character.

The local Steinberg representation is the quotient of locally constant functions on the projective line by constant functions, with the action induced from GL2. In the unnormalized convention, $V(1)$ fits into $0\to 1\to V(1)\to \mathrm{St}\to 0$. Its scalar endomorphism algebra obstructs a splitting. This local infinite-dimensional representation differs from the finite-field Steinberg representation.

For the standard representation of the symmetric group $S_3$, the unramified local factor of an Artin L-function, with $t=p^{-s}$, is respectively the displayed factor for identity, transposition, and three-cycle Frobenius cycle types. These follow by removing the trivial summand from the three-point permutation representation. If the inertia group contains a three-cycle, its invariant subspace in the standard representation is zero, so the ramified local factor is one.

In dimension two, $z\mapsto z^k$ for positive $k$ has local degree $k$ at zero, and $z\mapsto {\overline{z}}^{|k|}$ has degree $k$ for negative $k$. The proper map $z\mapsto |z{|}^2\in \mathbb{C}$ has isolated zero fiber and local degree zero, since its image misses every nearby nonreal target. These proper maps extend to the two-sphere. Taking their product with the identity of ${\mathbb{R}}^{n-2}$ realizes the same local degrees in every dimension $n\ge 2$. In dimension one an isolated local degree is restricted to $0,1,-1$.

Its fiber at $x$ is $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$. Transport changes its generator by the sign of local orientation along the path. For a connected nonorientable manifold, its degree-zero cohomology is zero, because no nonzero integer is invariant under sign reversal, whereas its degree-zero homology is $\mathbb{Z}/2$, the coinvariants under that reversal.

The list-chromatic index is the least integer $q$ such that every assignment of at least $q$ allowed colors to each edge admits a list edge coloring. Uniform identical lists show it is at least the chromatic index. The Galvin theorem for bipartite list edge coloring gives equality for bipartite graphs.

For a Lagrangian submanifold $L$ in an exact symplectic manifold with primitive $\lambda$, its period subgroup is $\{{\int }_c\lambda :c\in H_1(L;\mathbb{Z})\}\subset \mathbb{R}$. It is rational when this subgroup is discrete. If it is nonzero, it is $\gamma \mathbb{Z}$ for a unique $\gamma >0$. Some conventions reserve the word rational for this nonzero case. The zero subgroup corresponds to an exact differential form $\lambda {|}_L$; its least positive period is interpreted as $+\infty$ if that endpoint convention is admitted.

For a smooth nonconstant holomorphic disk $h$ in ${\mathbb{C}}^n$ with boundary on a Lagrangian submanifold $L$, its symplectic area is ${\int }_{D^2}|{\partial }_{s}h{|}^{2}dsdt>0$. The Cauchy-Riemann equations give ${\partial }_{t}h=J{\partial }_{s}h$, so this is the pullback of the standard symplectic form. The Stokes theorem identifies the area with the period of a primitive along the boundary loop. Thus the Liouville class of a Lagrangian submanifold evaluates nontrivially on this loop and cannot vanish.

The logit link maps a mean probability to its log odds: $g(p)=\log [p/(1-p)]$, $0<p<1$. Its inverse is $p=(1+e^{-\eta }{)}^{-1}$, used in logistic regression.

The identity link function equates the response mean with the linear predictor. In a generalized linear model it gives ${\mu }_i=x_i^T\beta$. It is the canonical link function for the normal distribution. For response families whose means must remain positive or between zero and one, using the identity link requires corresponding restrictions on the linear predictor.

For $y_{tt}+2k{y}_t=c^2{y}_{xx}$ with fixed endpoints, initial displacement zero and initial velocity a Dirac delta at $a\in (0,l)$, the Fourier sine series coefficient of that velocity is $2\sin (n\pi a/l)/l$. Each coefficient evolves by $T_n^{\prime \prime }+2k{T}_n^{\prime }+(n\pi c/l{)}^2{T}_n=0$. Here ${\omega }_n^2=(n\pi c/l{)}^2-k^2$. At critical damping, replace $\sin ({\omega }_{n}t)/{\omega }_n$ by $t$; for an overdamped mode replace it by $\sinh (\sqrt{k^2-(n\pi c/l{)}^2}t)/\sqrt{k^2-(n\pi c/l{)}^2}$. The initial conditions hold in the sense of distributions.

For nonresonant incompressible normal modes of a uniform-density shear flow with aligned uniform magnetic field, let $S=\sigma +k_{x}U(z)$, $k^2=k_x^2+k_y^2$ and $f=w/S$. The normal displacement is $f/i$. The linearized horizontal equations and incompressibility give $i{k}^2{p}_1=\rho (S{w}^{\prime }-S^{\prime }w)=\rho S^2{f}^{\prime }$. Substituting the ideal magnetohydrodynamic induction equation into normal momentum gives the displayed conservative equation. Across a zero-thickness shear layer, continuity of $f$ and of $(\rho S^2-k_x^2{B}^2)f^{\prime }$ supplies the interface conditions. They express common normal displacement and continuous perturbed magnetohydrodynamic total pressure, avoiding undefined products of distributional shear gradients.

Pair $2n$ linearly ordered endpoints uniformly. Each pair is a chord; its larger endpoint ends a vertex block, with all endpoints after the previous right endpoint merged into that vertex. Orient the chord from its right-endpoint block to its left-endpoint block. Removing the last endpoint and its mate leaves a uniform smaller pairing. Reinserting the mate into one of $2n-1$ equally likely gaps chooses an existing block proportionally to its endpoint count, or the final gap creates a self-loop. This proves equivalence with the linearized chord diagram model.

An equivalent continuous representation draws $n$ independent pairs of independent uniform points on $[0,1]$, orders the pairs by their maxima $R_1<\cdots <R_n$, and calls their minima $L_i$. The $R_i^2$ are uniform order statistics; conditional on all the $R_i$, the $L_i$ are independent uniform points on $[0,R_i]$.

In a uniform linearized chord diagram, $\mathrm{Pr}(\sqrt{n}{R}_1\ge x)=(1-x^2/n{)}^n\to e^{-x^2}$. Conditional on the right endpoints, $d_1(n)=2+{\sum }_{i=2}^n{1}_{\{L_i\le R_1\}}$, with independent success probabilities $R_1/R_i$. A Chernoff bound gives $R_i=(1+o(1))\sqrt{i/n}$ uniformly for $i\ge n^{1/10}$. The early indicators are $o(\sqrt{n})$, and the remaining conditional mean is $(2+o(1))n{R}_1$. Their conditional variance is $O_{\mathbb{P}}(\sqrt{n})$. Hence $d_1(n)/\sqrt{n}-2\sqrt{n}{R}_1\to 0$ in probability, proving the displayed tail limit. A self-loop's contribution is included in the constant two.

The relaxation modulus is the shear-stress response at lag s to a unit small step of shear strain. By linear superposition it is the causal memory kernel multiplying shear rate. Its decaying part represents stored stress that relaxes after deformation ceases.

For the convention $e^{i\omega t}$, complex viscosity is the ratio of oscillatory shear stress amplitude to shear rate amplitude. A causal relaxation modulus gives the displayed transform. Its associated complex shear modulus is $G^{\ast }=i\omega {\eta }^{\ast }=G^{\prime }+i{G}^{\prime \prime }$, where $G^{\prime }$ is the storage modulus and $G^{\prime \prime }$ the loss modulus.

A small step of shear stress $s_{0}H(t)$ produces shear strain $\gamma (t)=s_{0}J(t)$ from a relaxed state. This time-dependent deformation under held stress is creep. The causal constitutive convolution gives $\hat{s}=p\hat{G}\hat{\gamma }$, proving the displayed Laplace transform relation. For a Maxwell fluid, $J(t)=1/G_0+t/(G_0\tau )$: an elastic jump followed by viscous flow.

For $x^{\prime }-Ax=e^{\lambda t}v$, substituting an exponential particular solution reduces the differential equation to $(\lambda I-A)u=v$. The displayed solution exists whenever $\lambda$ is not an eigenvalue of $A$. Resonance can require polynomial factors in time. The general solution adds $e^{tA}c$, expressed using the matrix exponential.

Minimizing $\Vert y-X\theta {\Vert }^2$ gives the displayed equations by differentiation. Equivalently, the regression residual $r=y-X\hat{\theta }$ satisfies $X^{T}r=0$: it is orthogonal to every column of the design matrix. If $X$ has full column rank, the minimizer is unique and the fitted values are the orthogonal projection of $y$ onto its column space. A rank-deficient design matrix can give multiple coefficient vectors, while the projected fitted vector is still unique.

For an intercept and one predictor, put $S_{xx}={\sum }_i(x_i-\bar{x}{)}^2$ and $S_{xy}={\sum }_i(x_i-\bar{x})(y_i-\bar{y})$. The equations give $\hat{\beta }=S_{xy}/S_{xx}$ and $\hat{\alpha }=\bar{y}-\hat{\beta }\bar{x}$, provided $S_{xx}>0$. In the five-point example in linear regression, $\bar{x}=2$, $\bar{y}=2.8$, $S_{xx}=10$ and $S_{xy}=10$, giving $\hat{\beta }=1$ and $\hat{\alpha }=0.8$.

The normal equations describe the mathematical optimum, but explicitly forming and inverting $X^{T}X$ can amplify numerical problems. A QR decomposition computes the same least-squares solution through orthogonal coordinates; a singular value decomposition also exposes nearly dependent columns. This connects statistical fitting to numerical linear algebra and the choice of a stable algorithm. ^[ref]